Theoretical and finite element modeling of piezoelectric nanobeams with surface and flexoelectricity effects

In this paper, a size-dependent Euler-Bernoulli beam model, which takes the flexoelectricity, piezoelectricity, and dielectricity as well as the surface elasticity into consideration, is established. Theoretical solutions for the static bending deflection of thin beams under different loading (uniformly distributed and concentrated load) and boundary conditions (cantilevered, both ends simply supported, clamped-clamped), are established. Moreover, an iterative finite element algorithm is developed for the analysis of thin flexoelectric beams under any load and boundary conditions. Our numerical results show the direct bulk flexoelectricity always play the role of stiffening the beams with various boundary conditions, while the residual surface stresses behave either stiffening or softening the nanobeams dependent on boundary conditions. The present investigation also demonstrates the size-dependent effects of flexoelectricity on the bending rigidity and piezoelectricity.